Area of a Sector

A sector is a portion of a circle bounded by two radii and the arc joining their extremities. It is thus a form of a triangle with a covered base. In the figure, the portion AOB is a sector. Arc AB is called the arc of the sector and \angle AOB is called the angle of the sector.


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The area of a sector of a circle is equal to a fraction of the area of the circle determined by dividing the size of the angle by {360^ \circ }. So, if the angle of the sector is {90^ \circ }, the area of the sector is \frac{{90}}{{360}} or \frac{1}{4} the area of the circle. Thus, the area of a sector of the circle divided by the area of the whole circle is proportional to the angle of the sector divided by {360^ \circ }.

If the angle \angle AOB is given in degrees, say{N^ \circ }, then

Area of the sector: A = \frac{{\pi {r^2}}}{{360}} \times {N^ \circ }

Length of the arc: l = \frac{{2\pi r}}{{360}} \times {N^ \circ }

If the angle \angle AOB is given in radians, say \theta radians, then

Area of the sector, A = \frac{{\pi {r^2}}}{{360}} \times {\left( {\frac{{180}}{\pi } \times \theta } \right)^ \circ } = \frac{1}{2}{r^2}\theta

If the length l of the arc and the radius r of the circle are given, then

Area of the sector, A = \frac{1}{2}{r^2} \times \theta = \frac{1}{2}{r^2} \times \frac{1}{l}   where \theta = \frac{l}{r}

A = \frac{1}{2}r \times l

 

Example:

Find the area of a sector of {60^ \circ } in a circle of radius 10cm.

 

Solution:

The sector AOB has angle {60^ \circ }.


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            Its area is equal to \frac{{60}}{{360}} \times \pi {r^2}
Since the radius  = 10cm
Therefore A = \frac{1}{6} \times 3.1416 \times {(10)^2} = 52.36 square

 

Example:

The minute hand of a clock is 12cw long. Find the area which is described on the clock face between 6A.M to 6.20A.M.

 

Solution:

Given that the length of the minute hand's radius is  = 12cm
\therefore 60minutes  = {360^ \circ }

\therefore 20minutes  = {120^ \circ }

Since the area of sector  = \frac{{\pi {r^2}}}{{360}} \times {N^ \circ }

 = \frac{{3.1416 \times {{(12)}^2}}}{{360}} \times {120^ \circ }

 = \frac{1}{3} \times 3.1416 \times 144 = 150.7 square cm